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We study the $K$-user, $M$-subchannel parallel multiple-input-single-output(MISO) broadcast channel (BC) under arbitrary levels of partial channel stateinformation at the transmitter (CSIT). We show that the parallel subchannelsconstituting this setting are separable from a degrees-of-freedom (DoF) regionperspective if and only if the partial CSIT pattern is totally ordered. Thistotal order condition corresponds to users abiding by the same order, withrespect to their CSIT quality levels, in each of the parallel subchannels. Forinstance, let $\alpha_{k}^{[l]}$ and $\alpha_{j}^{[l]}$ be the CSIT qualityparameters for users $k$ and $j$ over subchannel $l$. Under total order, having$\alpha_{k}^{[l]} \geq \alpha_{j}^{[l]}$ implies that $\alpha_{k}^{[m]} \geq\alpha_{j}^{[m]}$ holds for every subchannel $m$. In this case, the entire DoFregion is achievable using simple separate coding, where asingle-subchannel-type transmission scheme is employed in each subchannel. Toshow this separability result, we first derive an outer bound for the DoFregion by extending the aligned image sets approach of Davoodi and Jafar to theconsidered setting. We then show that this outer bound coincides with the innerbound achieved through separate coding, given by the Minkowski sum of $M$single-subchannel DoF regions, under the total order condition, hence settlingthe if part of the main theorem. To prove the only if part of the theorem, weidentify a set of DoF tuples achievable through joint coding acrosssubchannels, yet not achievable through separate coding whenever the totalorder condition is violated. Moreover, we also highlight the implications ofour main result on the design of CSIT feedback schemes for multi-carriermulti-antenna wireless networks.

On the Separability of Parallel MISO Broadcast Channels Under Partial CSIT: A Degrees of Freedom Region Perspective